# Introduction to Macromolecular X-ray Crystallography

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Introduction to Macromolecular X-ray Crystallography
Biochem 300 Borden Lacy Print and online resources: Introduction to Macromolecular X-ray Crystallography, by Alexander McPherson Crystallography Made Crystal Clear, by Gale Rhodes Online tutorial with interactive applets and quizzes. Nice pictures demonstrating Fourier transforms Cool movies demonstrating key points about diffraction, resolution, data quality, and refinement. Notes from a macromolecular crystallography course taught in Cambridge

Overview of X-ray Crystallography
Crystal -> Diffraction pattern -> Electron density -> Model Resolution, Fourier transforms, the ‘phase problem’, B-factors, R-factors, R-free …

Diffraction: The interference caused by an object in the path of waves
(sound, water, light, radio, electrons, neutron..) Observable when object size similar to wavelength. Object Visible light: nm X-rays: nm, 1-2 Å

Can we image a molecule with X-rays?
Not currently. 1) We do not have a lens to focus X-rays. Measure the direction and strength of the diffracted X-rays and calculate the image mathematically. 2) The X-ray scattering from a single molecule is weak. Amplify the signal with a crystal - an array of ordered molecules in identical orientations.

Varying intensity Reciprocal space Resolution What is the physical basis for the pattern you see? How do you get electron density from this information?

The wave nature of light
f(x) = Fcos2π(ux + a) f(x) = Fsin2π(ux + a) x F = amplitude u = frequency a = phase ul=c f(x) = cos 2πx f(x) = 3cos2πx f(x) = cos2π(3x) f(x) = cos2π(x + 1/4)

Interference of two waves
Wave 1 + Wave 2 Wave 1 Wave 2 In-phase Out -of-phase

Bragg’s Law Sin q = AB/d AB = d sin q AB + BC = 2d sin q nl = 2d sin q

nl = 2d sin q

We are sampling the continuous transform at specific points determined by the periodic lattice.
The lattice determines the spacing in the diffraction pattern. The intensity of the spots contains the information about the lattice content. Each individual spot on the diffraction pattern contains information about your entire molecule.

Diffraction pattern The intensity of each spot contains
Practically: Assign a coordinate (h, k, l) and intensity (I) to every spot in the diffraction pattern— Index and Integrate. Ihkl , shkl The intensity of each spot contains information about the entire molecule. The spacing of the spots is due to the size and symmetry of your lattice.

Fourier transform: F(h)= ∫ f(x)e2πi(hx)dx where units of h are reciprocals of the units of x Reversible! f(x)= ∫ F(h)e-2πi(hx)dh

Calculating an electron density function from the diffraction pattern
F(h) = Fcos2π(uh+ a) F(h) = Fsin2π(uh + a) r(x) = ∫ F(h)e-2πi(hx)dh F = amplitude u = frequency a = phase Experimental measurements: Ihkl, shkl Fhkl ~ √Ihkl

Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous Replacement) Anomalous Scattering Methods Molecular Replacement Methods Direct Methods

Heavy Atom Methods (Isomorphous Replacement)
The unknown phase of a wave of measurable amplitude can be determined by ‘beating’ it against a reference wave of known phase and amplitude. Combined Wave Unknown Reference

Generation of a reference wave:
Max Perutz showed ~1950 that a reference wave could be created through the binding of heavy atoms. Heavy atoms are electron-rich. If you can specifically incorporate a heavy atom into your crystal without destroying it, you can use the resulting scatter as your reference wave. Crystals are ~50% solvent. Reactive heavy atom compounds can enter by diffusion. Derivatized crystals need to be isomorphous to the native.

Native Fnat Heavy atom derivative Fderiv

The steps of the isomorphous replacement method

Heavy Atom Methods (Isomorphous Replacement)
The unknown phase of a wave of measurable amplitude can be determined by ‘beating’ it against a reference wave of known phase and amplitude. FPH FP FH and aH Can use the reference wave to infer aP. Will be either of two possibilities. To distinguish you need a second reference wave. Therefore, the technique is referred to as Multiple Isomorphous Replacement (MIR).

Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous Replacement) Anomalous Scattering Methods Molecular Replacement Methods Direct Methods

Anomalous scattering Incident X-rays can resonate with atomic electrons to result in absorption and re-emission of X-rays. Results in measurable differences in amplitude Fhkl ≠ F-h-k-l

Use of synchrotron radiation allows one to ‘tune’ the wavelength of the X-ray beam to the absorption edge of the heavy atom. Incorporation of seleno-methionine into protein crystals.

Anomalous scattering/dispersion in practice
Anomalous differences can improve the phases in a MIR experiment (MIRAS) or resolve the phase ambiguity from a single derivative allowing for SIRAS. Measuring anomalous differences at 2 or more wavelengths around the absorption edge: Multiple-wavelength anomalous dispersion (MAD). Advantage: All data can be collected from a single crystal. Single-wavelength anomalous dispersion (SAD) methods can work if additional phase information can be obtained from density modification.

Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous Replacement) Anomalous Scattering Methods Molecular Replacement Methods Direct Methods

Molecular Replacement
If a model of your molecule (or a structural homolog) exists, initial phases can be calculated by putting the known model into the unit cell of your new molecule. 1- Compute the diffraction pattern for your model. 2- Use Patterson methods to compare the calculated and measured diffraction patterns. 3- Use the rotational and translational relationships to orient the model in your unit cell. 4- Use the coordinates to calculate phases for the measured amplitudes. 5- Cycles of model building and refinement to remove phase bias.

Direct Methods Ab initio methods for solving the phase problem either by finding mathematical relationships among certain phase combinations or by generating phases at random. Typically requires high resolution (~1 Å) and a small number of atoms. Can be helpful in locating large numbers of seleno-methionines for a MAD/SAD experiment.

Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous Replacement) Anomalous Scattering Methods Molecular Replacement Methods Direct Methods F = amplitude u = frequency a = phase FT r(x,y,z) electron density

Electron density maps

Are phases important? Duck intensities and cat phases

Does molecular replacement introduce model bias?
Cat intensities with Manx phases

An iterative cycle of phase improvement
Building Refinement Solvent flattening NCS averaging

Model building Interactive graphics programs allow for the creation of a ‘PDB’ file. Atom type, x, y, z, Occupancy, B-factor

The PDB File: ATOM 1 N GLU A 27 41.211 44.533 94.570 1.00 85.98
ATOM CA GLU A ATOM C GLU A ATOM O GLU A ATOM CB GLU A ATOM CG GLU A ATOM CD GLU A ATOM OE1 GLU A ATOM OE2 GLU A ATOM N ARG A ATOM CA ARG A ATOM C ARG A ATOM O ARG A ATOM CB ARG A ATOM CG ARG A ATOM CD ARG A ATOM NE ARG A ATOM CZ ARG A ATOM NH1 ARG A ATOM NH2 ARG A .

Occupancy What fraction of the molecules have an atom at this x,y,z position? B-factor How much does the atom oscillate around the x,y,z position? Can refine for the whole molecule, individual sidechains, or individual atoms. With sufficient data anisotropic B-factors can be refined.

Least -squares refinement
= S whkl (|Fo| - |Fc|)2hkl Apply constraints (ex. set occupancy = 1) and restraints (ex. specify a range of values for bond lengths and angles) Energetic refinements include restraints on conformational energies, H-bonds, etc. Refinement with molecular dynamics An energetic minimization in which the agreement between measured and calculated data is included as an energy term. Simulated annealing often increases the radius of convergence.

R = Monitoring refinement S||Fobs| - |Fcalc|| S|Fobs|
Rfree: an R-factor calculated from a test set that has not been used in refinement.